Finite dimensional global attractor for a semi-discrete fractional nonlinear Schrödinger equation

Caterina Calgaro (1, 2) , Olivier Goubet (3) , Ezzeddine Zahrouni (4)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
4 Faculté des sciences économiques de Nabeul

We consider a semi-discrete in time Crank-Nicolson scheme to discretise a weakly damped forced nonlinear fractional Schrödinger equation u t − i(−∆) α u + i|u| 2 u + γu = f for α ∈ (1 2 , 1) considered in the the whole space R. We prove that such semi-discrete equation provides a discrete infinite dimensional dynamical in H α (R) that possesses a global attractor in H α (R). We show also that if the external force is in a suitable weighted Lebesgue space then this global attractor has a finite fractal dimension.

Mots clés: fractal dimension; Nonlinear Schrödinger equations; Crank-Nicolson scheme; global attractor

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques [math]

Caterina Calgaro, Olivier Goubet, Ezzeddine Zahrouni. Finite dimensional global attractor for a semi-discrete fractional nonlinear Schrödinger equation. Mathematical Methods in the Applied Sciences, 2017, ⟨10.1002/mma.4409⟩. ⟨hal-01388788⟩